Mega Man X4

It's generally not recommended to begin the eight bosses with Web Spider - go for Magma Dragoon first, as you can get to his fight with reinforcements if you're smart.

Battling Web Spider requires you to be fast, as he only shows up for a few moments before shooting off that semi-homing Lightning Web of his and retreating. Slash him once as he goes down, jump up and slash him again. Turn around then dash, followed quickly by a jump (be sure to have plenty of room between Web Spider and the wall so you can do this dodge). If you're feeling really lucky, you can jump over the Lightning Web and smack Spider one more time before he goes up, however this is risky. If not, you can just dash to the wall, jump up to it and dash+jump back towards the middle of the stage.

When Spider does the big web thing, it's a matter of just tracking him down and slashing him to bits. Just don't forget to dodge the Lightning Web and you'll be fine.